I learned an interesting fact about Fibonacci numbers recently while watching a lecture on number theory. Fibonacci numbers can be used to approximately convert from miles to kilometers and back.

Here is how.

Take two consecutive Fibonacci numbers, for example 5 and 8. And you're done converting. No kidding – there are 8 kilometers in 5 miles. To convert back just read the result from the other end - there are 5 miles in 8 km!

Another example. Let's take the consecutive Fibonacci numbers 21 and 34. What this tells us is that there are approximately 34 km in 21 miles and vice versa. (The exact answer is 33.79 km.)

If you need to convert a number that is not a Fibonacci number, just express the original number as a sum of Fibonacci numbers and do the conversion for each Fibonacci number separately.

For example, how many kilometers are there in 100 miles? Number 100 can be expressed as a sum of Fibonacci numbers 89 + 8 + 3. Now, the Fibonacci number following 89 is 144, the Fibonacci number following 8 is 13 and the Fibonacci number following 3 is 5. Therefore the answer is 144 + 13 + 5 = 162 kilometers in 100 miles. This is less than 1% off from the precise answer, which is 160.93 km.

Another example, how many miles are there in 400 km? Well, 400 is 377 + 21 + 2. Since we are going the opposite way now from miles to km, we need the preceding Fibonacci numbers. They are 233, 13 and 1. Therefore there are 233 + 13 + 1 = 247 miles in 400 km. (The correct answer is 248.55 miles.)

Just remember that if you need to convert from km to miles, you need to find the preceding Fibonacci number. But if you need to convert from miles to km, you need the subsequent Fibonacci number.

If the distance you're converting can be expressed as a single Fibonacci number, then for numbers greater than 21 the error is always around 0.5%. However, if the distance needs to be composed as a sum of n Fibonacci numbers, then the error will be around sqrt(n)·0.5%.

Here's why it works.

Fibonacci numbers have a property that the ratio of two consecutive numbers tends to the Golden ratio as numbers get bigger and bigger. The Golden ratio is a number and it happens to be approximately 1.618.

Coincidentally, there are 1.609 kilometers in a mile, which is within 0.5% of the Golden ratio.

Now that we know these two key facts, we can figure out how to do the conversion. If we take two consecutive Fibonacci numbers, Fn+1 and Fn, we know that their ratio Fn+1/Fn is approximately 1.618. Since the ratio is also almost the same as kilometers per mile, we can write Fn+1/Fn = [mile]/[km]. It follows that Fn·[mile] = Fn+1·[km], which translates to English as "n-th Fibonacci number in miles is the same as (n+1)-th Fibonacci number in kilometers".

That's all there is to it. A pure coincidence that the Golden ratio is almost the same as kilometers in a mile.

klabr,
I think it must be a coincidence. The length of a meter was originally based on the circumference of the Earth, while the length of a mile was based on the Roman pace. As coincidences go, this is a pretty strange and interesting one.

You could also convert from feet to yards by multiplying it by the speed of light and dividing by a 100 million!
That's all there is to it. The speed of light, a ratio which occurs so often in physics equations, can also be used inaccurately to convert between feet and yards.

No, y'all wisely abandoned them when a better system came along. The fact that we Americans stubbornly clung to them (and continue to cling to them) may make it look like you ceded them to us, but in reality it's no one's fault but our own.

Proof's really easy: it's true for n=1. Now suppose it's true for all n<=k. If n+1 is Fibonacci number, we are done. Otherwise, this number n+1 is bigger than some Fibonacci number Fi. Now let's look at the number n+1-Fi. As this number is smaller than n+1, then according to the inductive hypothesis it can be expressed as a sum of Fibonacci numbers. Therefore n+1 can be expressed as Fi + (n+1-Fi), where Fi is a Fibonacci number and (n+1-Fi) is a sum of Fibonacci numbers.

Um. Um. Yeah, I don't even know what Fibonacci numbers are! High school student, I feel like I should PROBABLY know what that is... but nope, I got nothing. I'll bookmark the page since I can never get conversions right, though... =w=

We know that any natural number can be expressed (not uniquely) as the sum of Fibonacci factors. Given a (potentially huge) natural number, how can you write it as the sum of Fibonacci factors so that the smallest factor is as large as possible?

@Nathan, i was thinking the same thing. I would start with the list of all Fibonacci numbers less than x (your number to convert). say F(n) is the largest of these. Now add smaller numbers to F(n) starting with F(n-1). If your total ever goes above x, then remove the last number you added and decrement n. stop when your sum equals x. The numbers you've added are the factors.

Hmmm... I have lived with both systems of measurement my whole life. I have rarely had to do a conversion. Use both long enough and you will get a feel for it. Many people I know can use them interchangeably. Generally, I am not worried by being precise to the tenths, hundredths, or thousandths. That may be required for the Olympic judges or NASA engineers, but not in every day life. I know what driving at 100kms/hr feels like and I know what 100mi/hr feels like. (On most highways I will get a ticket doing the second one.) It's similar to learning to speak a second or third language. Many times there are no exact translations in the alternate language but you sometimes come close.

Of course everyone is quite right with "It's easier to Goggle or use my calculator", however this is a handy way to quickly approximate figures in your head.

Another handy way for programmers (like I once was) is to use hexadecimal to decimal conversions. Having dealt with the large round numbers daily, they become ingrained and 30hex = 48dec or 100dec = 64hex is instantly retrievable. If we say that km = decimal and that miles = hexadecimal, the (approximate!) conversions are easy.

I had to have a system such as this as I have an imported sports car which I refused to mutilate by changing the speedo. e.g. if my km speedo tells me 50kph, I know it's around 30mph, if it said 100kph, that's roughly 64mph.

For all the naysayers, the author is just trying to prove a theory.. We all (including the brilliant author) know that 1.6xKM = miles, but this is a pretty intriguing observation.. So just shup up, shove it, and read and digest the fact. And if you cannot come up with awesome ideas like these, too bad for you losers.. Well done Peter..

Since the ratio of the Fibonacci numbers approximates the golden ratio as the numbers get bigger, you can just memorize a few of the bigger ones and move the decimals where they're needed. Instead of using 1 <-> 2, you can use 102,334,155 <-> 165,580,141, so 1.02 <-> 1.67, or 2,504,730,781,961 <-> 4,052,739,537,881, so 25.0 <-> 40.5.